Prove the following statement: A set K in the topology X is compact if and only

Question

Prove the following statement: A set K in the topology X is compact if and only if for any collection C of closed subsets of K, if any finite subsets of C intersect nontrivially, then all of C intersects nontrivially.

Explanation / Answer

This proof is not an easy read. Well you have to have in mind thelogical form... at least i have trouble with it... the secondproperty is stated as the finite intersection property for closedcentered systems. Notation: for inclusion, [] for intersection, C(A) forcomplement of A respect to K, U for union, i will put UGn, nr). TakeC a family of closed sets, and K to be the space. Then p is Kis compact, q is Any finite subcollection of C has finiteintersection, and r is The intersection of [] C is nonempty. ->   We will assume compacity and prove that itimplies the contrapositive statement no r -> no q. This is the same, compacity implies the other thing. Take then that {G_i} is a collection of open sets that covers K andK is compact. Provided that C_i = K - G_i, we have that [] C_i =emptyset... because K U G_i. Then because of compacity,there is a finite covering {G_n} n

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.